Finite Group Databasebeta

← #199157

Subgroups — #199157

Order 512.

1199 conjugacy classes of subgroups.

The subgroup lattice has 489 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3206C211yesmaximal, characteristic
2#34371C211yesmaximal, characteristic, complemented
3#21816C211yesmaximal, characteristic
4#17746C211yesmaximal, characteristic, complemented
5#17589C211yesmaximal, characteristic
6#9301C218yesmaximal, complemented
7#7956C211yesmaximal, characteristic, complemented
8#7858C211yesmaximal, characteristic, complemented
9C2×#532C2211yescharacteristic
10C22×C4⋊Q8C2211yescharacteristic
11#3206—22no—
12#3206—22no—
13#3206C414yescomplemented
14#3206C2214yes—
15C2×#492C412yescomplemented
16C2×#487C412yescomplemented
17C2×#461—22no—
18C2×#448—22no—
19#2920C2211yescharacteristic
20#2920C2211yescharacteristic
21C2×#431C2211yescharacteristic
22#2381—22no—
23#2365—22no—
24#2345—22no—
25Q8×C22⋊C4—22no—
26#2331C2214yescomplemented
27#2327—22no—
28#2313—22no—
29#2283C2211yescharacteristic
30#2283C2211yescharacteristic
31#1980—22no—
32#1963C2214yes—
33#1789—22no—
34#1558—22no—
35#1548—22no—
36#1548C2214yescomplemented
37#1517C2214yes—
38#1510—22no—
39#1505—22no—
40#1286C2214yescomplemented
41#1274C2214yescomplemented
42C22×#41—21no—
43C23×Q8D412yes—
44C22×SD16—24no—
45#532—21no—
46#532—21no—
47#532—21no—
48#532—21no—
49#532—21no—
50#532—21no—
51#532C2×C412yes—
52#532C2312yes—
53C2×C4⋊Q8—21no—
54C2×C4⋊Q8—21no—
55C2×C4⋊Q8—21no—
56C2×C4⋊Q8—21no—
57C2×C4⋊Q8—21no—
58C2×C4⋊Q8—21no—
59C2×C4⋊Q8C2×C411yescharacteristic
60C2×C4⋊Q8C2×C412yes—
61C2×C4⋊Q8C2×C412yes—
62C2×C4⋊Q8D412yes—
63C2×C4⋊Q8D412yes—
64C2×C4⋊Q8D412yes—
65C2×C4⋊Q8C2312yes—
66C2×#127C2×C411yescharacteristic
67C2×#124—21no—
68C2×#121—21no—
69C2×#121—21no—
70C2×C4×Q8—24no—
71C2×C4×D4—21no—
72C2×C4×D4—21no—
73C22×C4⋊C4—21no—
74C22×C4⋊C4—21no—
75C22×C42C2311yescharacteristic
76#492—24no—
77#492C2×C418yes—
78#487—24no—
79#487C2×C418yes—
80#461—28no—
81#461—28no—
82#448—28no—
83#448—28no—
84#432—21no—
85#432—21no—
86#432—21no—
87#432—21no—
88#431—21no—
89#431—21no—
90#431—21no—
91#431—21no—
92#431—21no—
93#431C2×C412yes—
94#431C2312yes—
95#431C2312yes—
96#423—21no—
97#423C2312yes—
98C2×C4⋊C8—21no—
99C2×#104—28no—
100C2×#103—24no—
101#407—21no—
102#407—21no—
103C2×C22⋊C8—21no—
104C4×M16—21no—
105C4×M16—21no—
106C2×C4×C8C2×C412yes—
107#391—24no—
108#391—28no—
109#389—24no—
110#387—28no—
111#387—28no—
112#387—28no—
113#383—21no—
114#383—21no—
115#382—21no—
116#382C2312yes—
117#381—21no—
118#381D412yes—
119#380—24no—
120#380—21no—
121#380—21no—
122C4×C4⋊C4—24no—
123C4×C4⋊C4—21no—
124C4×C4⋊C4D412yes—
125C4×C22⋊C4—24no—
126C4×C22⋊C4—21no—
127C4×C22⋊C4D412yes—
128#376D412yes—
129#376C2312yes—
130#333—28no—
131#327—28no—
132C2×#41—22no—
133C2×#41—22no—
134C2×#41—41no—
135C2×#41—41no—
136C2×#41—84no—
137C22×Q8—22no—
138C22×Q8—22no—
139C22×Q8—22no—
140C22×Q8—21no—
141C22×Q8—84no—
142C22×Q8C22⋊C414yes—
143C22×Q8C22⋊C414yes—
144C22×Q8C2×D414yes—
145C22×D4—21no—
146C22×D4—21no—
147C22×D4—21no—
148C22×D4—41no—
149C23×C4—22no—
150C23×C4—41no—
151C23×C4C2×D411yescharacteristic
152C23×C4C2×D411yescharacteristic
153C23×C4C2×D411yescharacteristic
154C2×SD16—216no—
155C2×SD16—88no—
156C2×M16—22no—
157C2×M16—41no—
158C22×C8—22no—
159C4⋊Q8—21no—
160C4⋊Q8—22no—
161C4⋊Q8—22no—
162C4⋊Q8—22no—
163C4⋊Q8—22no—
164C4⋊Q8—21no—
165C4⋊Q8—21no—
166C4⋊Q8—22no—
167C4⋊Q8—22no—
168C4⋊Q8—22no—
169C4⋊Q8—22no—
170C4⋊Q8—22no—
171C4⋊Q8—22no—
172C4⋊Q8—22no—
173C4⋊Q8—22no—
174C4⋊Q8C22⋊C414yes—
175C4⋊Q8C22⋊C414yes—
176C4⋊Q8C22×C412yes—
177C4⋊Q8C22×C414yes—
178C4⋊Q8C2×D414yes—
179#127—21no—
180#127—21no—
181#127—21no—
182#127C22×C412yes—
183#124—22no—
184#124—22no—
185#124—41no—
186#124—41no—
187#121—22no—
188#121—22no—
189#121—22no—
190#121—22no—
191#121—41no—
192#121—41no—
193#121—41no—
194#121—41no—
195C4×Q8—88no—
196C4×D4—22no—
197C4×D4—22no—
198C4×D4—22no—
199C4×D4—22no—
200C4×D4—41no—
201C4×D4—41no—
202C4×D4—41no—
203C4×D4—41no—
204C2×C4⋊C4—22no—
205C2×C4⋊C4—22no—
206C2×C4⋊C4—22no—
207C2×C4⋊C4—22no—
208C2×C4⋊C4—22no—
209C2×C4⋊C4—22no—
210C2×C4⋊C4—22no—
211C2×C4⋊C4—22no—
212C2×C4⋊C4—22no—
213C2×C4⋊C4—22no—
214C2×C4⋊C4—22no—
215C2×C4⋊C4—22no—
216C2×C4⋊C4—21no—
217C2×C4⋊C4—21no—
218C2×C4⋊C4—22no—
219C2×C4⋊C4—22no—
220C2×C4⋊C4—28no—
221C2×C22⋊C4—22no—
222C2×C22⋊C4—22no—
223C2×C22⋊C4—41no—
224C2×C22⋊C4—41no—
225C2×C42—22no—
226C2×C42—28no—
227C2×C42C22×C411yescharacteristic
228C2×C42C22×C411yescharacteristic
229C2×C42C2×D414yes—
230C2×C42C2×D411yescharacteristic
231C2×C42C2×D411yescharacteristic
232C2×C42C2×D411yescharacteristic
233C2×C42C2×D411yescharacteristic
234C2×C42C2×D411yescharacteristic
235C2×C42#4111yescharacteristic
236C2×C42#4111yescharacteristic
237C2×C42#4111yescharacteristic
238C2×C42#4111yescharacteristic
239C2×C42C2411yescharacteristic
240C4⋊C8—22no—
241C4⋊C8—22no—
242C4⋊C8—22no—
243C4⋊C8—22no—
244C4⋊C8—22no—
245C4⋊C8—41no—
246#104—88no—
247#103—216no—
248C22⋊C8—22no—
249C22⋊C8—22no—
250C22⋊C8—22no—
251C22⋊C8—41no—
252C22⋊C8—41no—
253C22⋊C8—41no—
254#97—41no—
255#97—41no—
256C4×C8—22no—
257C4×C8—22no—
258C4×C8—41no—
259C4×C8C22×C414yes—
260#95—28no—
261#95—28no—
262#95—22no—
263#95—22no—
264#95—22no—
265#95—22no—
266#95—22no—
267#95—22no—
268#95—22no—
269#95C2×D414yes—
270C24—41no—
271C24#12011yescharacteristic
272#41—48no—
273#41—84no—
274#41—84no—
275#41—84no—
276C2×Q8—24no—
277C2×Q8—24no—
278C2×Q8—24no—
279C2×Q8—22no—
280C2×Q8—48no—
281C2×Q8—48no—
282C2×Q8—41no—
283C2×Q8—84no—
284C2×Q8—82no—
285C2×Q8—84no—
286C2×Q8—84no—
287C2×Q8—84no—
288C2×Q8C2×C22⋊C418yes—
289C2×D4—22no—
290C2×D4—22no—
291C2×D4—22no—
292C2×D4—44no—
293C2×D4—42no—
294C2×D4—42no—
295C2×D4—41no—
296C2×D4—41no—
297C2×D4—41no—
298C2×D4—82no—
299C2×D4—82no—
300C2×D4—84no—
301C22×C4—24no—
302C22×C4—24no—
303C22×C4—24no—
304C22×C4—24no—
305C22×C4—24no—
306C22×C4—24no—
307C22×C4—42no—
308C22×C4—42no—
309C22×C4—41no—
310C22×C4—42no—
311C22×C4—42no—
312C22×C4—84no—
313C22×C4—84no—
314C22×C4C2×C22⋊C411yescharacteristic
315C22×C4C2×C22⋊C411yescharacteristic
316C22×C4C4×D411yescharacteristic
317C22×C4C4×D411yescharacteristic
318C22×C4C4×D411yescharacteristic
319C22×C4C4×D411yescharacteristic
320C22×C4#12011yescharacteristic
321C22×C4#12011yescharacteristic
322C22×C4#12011yescharacteristic
323C22×C4#12111yescharacteristic
324C22×C4#12111yescharacteristic
325C22×C4#12111yescharacteristic
326C22×C4#12111yescharacteristic
327C22×C4#12311yescharacteristic
328C22×C4#12311yescharacteristic
329C22×C4C22×D411yescharacteristic
330C22×C4C22×D411yescharacteristic
331C22×C4C22×D411yescharacteristic
332SD16—816no—
333M16—44no—
334M16—82no—
335C2×C8—24no—
336C2×C8—44no—
337C2×C8—42no—
338C2×C8—42no—
339C4⋊C4—24no—
340C4⋊C4—24no—
341C4⋊C4—24no—
342C4⋊C4—24no—
343C4⋊C4—24no—
344C4⋊C4—24no—
345C4⋊C4—24no—
346C4⋊C4—24no—
347C4⋊C4—22no—
348C4⋊C4—22no—
349C4⋊C4—44no—
350C4⋊C4—44no—
351C4⋊C4—41no—
352C4⋊C4—41no—
353C4⋊C4—88no—
354C22⋊C4—44no—
355C22⋊C4—44no—
356C22⋊C4—42no—
357C22⋊C4—42no—
358C22⋊C4—42no—
359C22⋊C4—42no—
360C42—28no—
361C42—44no—
362C42—88no—
363C42C2×C22⋊C411yescharacteristic
364C42C2×C22⋊C411yescharacteristic
365C42C4×D411yescharacteristic
366C42C4×D411yescharacteristic
367C42C4×D411yescharacteristic
368C42C4×D411yescharacteristic
369C42#12111yescharacteristic
370C42#12111yescharacteristic
371C42#12111yescharacteristic
372C42#12111yescharacteristic
373C42#12311yescharacteristic
374C42#12311yescharacteristic
375C42C23×C411yescharacteristic
376C42C22×D411yescharacteristic
377C42C2×#4111yescharacteristic
378C42C2×#4111yescharacteristic
379C23—22no—
380C23—22no—
381C23—41no—
382C23—42no—
383C23—82no—
384C23—82no—
385C23—82no—
386C23#38611yescharacteristic
387C23#38611yescharacteristic
388C23C2×#12011yescharacteristic
389Q8—48no—
390Q8—48no—
391Q8—48no—
392Q8—44no—
393Q8—84no—
394Q8—82no—
395D4—44no—
396D4—44no—
397D4—82no—
398D4—82no—
399D4—82no—
400D4—84no—
401D4—84no—
402D4—82no—
403C2×C4—22no—
404C2×C4—22no—
405C2×C4—22no—
406C2×C4—22no—
407C2×C4—22no—
408C2×C4—22no—
409C2×C4—22no—
410C2×C4—22no—
411C2×C4—22no—
412C2×C4—22no—
413C2×C4—28no—
414C2×C4—41no—
415C2×C4—41no—
416C2×C4—41no—
417C2×C4—41no—
418C2×C4—41no—
419C2×C4—41no—
420C2×C4—41no—
421C2×C4—48no—
422C2×C4—44no—
423C2×C4—44no—
424C2×C4—44no—
425C2×C4—44no—
426C2×C4—44no—
427C2×C4—44no—
428C2×C4—42no—
429C2×C4—84no—
430C2×C4—84no—
431C2×C4—84no—
432C2×C4—82no—
433C2×C4—84no—
434C2×C4#38611yescharacteristic
435C2×C4#38611yescharacteristic
436C2×C4#38611yescharacteristic
437C2×C4#38611yescharacteristic
438C2×C4#38611yescharacteristic
439C2×C4#38611yescharacteristic
440C2×C4C22×C22⋊C411yescharacteristic
441C2×C4C2×C4×D411yescharacteristic
442C2×C4C2×C4×D411yescharacteristic
443C2×C4C2×#12111yescharacteristic
444C2×C4C2×#12111yescharacteristic
445C2×C4C2×#12311yescharacteristic
446C2×C4#57714yes—
447C8—44no—
448C8—82no—
449C8—82no—
450C22—21no—
451C22—21no—
452C22—21no—
453C22—21no—
454C22—21no—
455C22—21no—
456C22—21no—
457C22—44no—
458C22—82no—
459C22—82no—
460C22—82no—
461C22—164no—
462C22#190512yes—
463C22C2×#38611yescharacteristic
464C22#301012yes—
465C4—21no—
466C4—21no—
467C4—21no—
468C4—21no—
469C4—21no—
470C4—21no—
471C4—21no—
472C4—21no—
473C4—21no—
474C4—48no—
475C4—84no—
476C4—84no—
477C4—82no—
478C4#191112yes—
479C4#294912yes—
480C4#304312yes—
481C2—22no—
482C2—22no—
483C2—41no—
484C2—162no—
485C2#2093611yescharacteristic
486C2C2×#190511yescharacteristic
487C2C2×#191111yescharacteristic
subgroupG/H
center, socleC22C2×#386
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
prime radicalC2×#532C22
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. #34371C2semidirect
4. #17746C2semidirect
6. #9301C2semidirect
7. #7956C2semidirect
8. #7858C2semidirect
13. #3206C4semidirect
15. C2×#492C4semidirect
16. C2×#487C4semidirect
26. #2331C22semidirect
36. #1548C22semidirect
40. #1286C22semidirect
41. #1274C22semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C2×#32062
#343712
#218162
#177462
#175892
#93012
#79562
#78582